Is convex function? (related to sup.)

Question

$p(\alpha)= \sup_{x} xf(\alpha) \\ \quad \quad\quad \text{s. t.} \quad x \leq h(\alpha)$

where $f(\alpha)$ is convex function of $\alpha$ and $h(\alpha)$ is non convex function of $\alpha$.

Is $p(\alpha)$ convex function?

Answer 1

No. Take $f(x)=x^{2}, h(\alpha)=-1$. Then $p(\alpha)=-\alpha^{2}$ which is not convex.In general, for any non-negative convex function $f$, $p(\alpha)=f(\alpha) h(\alpha)$ and product of a non-negative convex function and an arbitrary function need not be convex.